918 Proceedings of Royal Society of Edinburgh. [sess. 
quand le produit k renfermera une seule fois chacune des 
lettres 
y , • • , y, 
la formnle (15) etant relative au cas oil Ton sera oblige 
d’operer entre ces lettres prises deux a deux un nombre pair 
d’echanges, pour passer du produit | a/3y . . . rj | au produit \k\” 
The obtaining of the results 
I a Py I = I fiy a I = I y a P 1 1 
= - | a.y/3 | = - |/3ay| = - | y/3a | j 
from transformations of the form 
1 a/5 | = ~ |/?a| 
is then shortly considered ; and this is followed by a concluding 
paragraph in which the statement occurs that “ cette decomposition 
(des sommes alternees en facteurs symboliques) une fois operee on 
peut s’en servir avec avantage pour decouvrir ou pour demontrer 
les principales proprietes des sommes alternees.” 
Cauchy’s position is thus seen to be very different from Grass- 
mann’s. Grassmann was not concerned with determinants : his 
problem was to solve the set of equations 
a Y x + a 2 y + = a 0 j 
\x + b 0 + b 3 z = \ j- 
«i* + <HH + c 3 2 = «o ) 
and he satisfied himself by a curious process of reasoning that 
x _ ( fl 0 + 5 0 + O q)(u 2 + b 2 + cf)(a 2 + b 3 + c s ) 
(a 1 + \ + c 1 )(a 2 + b 2 + c 2 )(a 3 + b s + c 3 ) ’ 
provided that the multiplications indicated be performed according 
to the laws of “ outer multiplication.” Cauchy, on the other 
hand, starts with the determinant formed from 
a 1 a 2 a. 
